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array: a set of objects in equal rows and equal

attributes: a characteristic or distinctive feature—

benchmark: a reference that is based on

categorical data: data that represents individuals

demonstrate fluency: demonstrating the ability

expression: a mathematical phrase that

Examples: 23 x 67; 33-.

even : a whole number that is divisible by 2.

features (of the data set): features include the

flipping/reflecting: a transformation creating a

fluency: refers to having efficient and accurate

identity property of multiplication: if you

line plot: a diagram showing frequency of data on

median (feature of data): when the numbers

mode (feature of data): the number that

model: to represent a mathematical situation with

multiple: the product of a whole number and any

other whole number.

number sentence: an equation or comparison.

numerical data: represent objects or individuals

odd: a whole number that is not divisible by 2.

outlier: a number in a set of data that is much

prism: a 3-dimensional figure in which all of the

range (feature of data): the difference

set: a collection of distinct elements or items. Math at hand: A mathematics handbook (p. 534).

sliding/translating: a transformation that involves

transformations: the mapping, or movement of

transforming shapes: changing plane figures

turning/rotating: a transformation that involves

associative property of addition: the sum

associative property of multiplication: the

attributes: a characteristic or distinctive feature—

categorical data: data that represents

center point (of rotation): the point that a

commonly used fractions: halves, thirds,

composite number: a number that has more

coordinate systems: two-dimensional systems

distributive property: when one of the factors of

even: a whole number that is divisible by 2. Math at hand: A mathematics handbook (p. 523).

expression: a mathematical phrase that

factor: an integer that will divide evenly

flipping/reflecting: a transformation creating a

fractions: a way of representing part of a whole

generalizations: reasoning about the structure of

model: to represent a mathematical situation with

net of a prism: a flat 2-dimensional shape that

number sentence: an equation or comparison.

numerical data: data that represent objects or

odd: a whole number that is not divisible by 2. Math at hand: A mathematics handbook (p. 529).

partitive: distribution division that involves

number of groups is known. Example: How would

prime number: a number that has exactly two

quotative: measurement division that involves

rotational symmetry: a property of a figure that

is mapped onto itself by a rotation of 1800 or less.

sliding/translating: a transformation that

square number: the number of dots in a square

transformation: the mapping, or movement of all

transforming shapes: changing plane figures by

translation: a transformation in which a figure is

turning/rotating: a transformation that involves

unit fraction: a fraction with a numerator of 1, for

associative property of addition: the sum

associative property of multiplication: the

benchmark: a reference that is based on

common factor: a number that is a factor of two

common multiple: a number that is a multiple of

composing or decomposing numbers:

conjecture: A proposition which is consistent with

to be false. It is synonymous with hypothesis.

corresponding angles: angles that are in the

corresponding sides of similar triangles:

distributive property: when one of the factors of

factor: an integer that will divide evenly

functions: relations in which every value of x has

a unique value of y.

image: a figure that is created after a shape

isometric representations: drawings that

linear (function) equation: an equation whose

mat plans: drawings of the base of a cube on

mean: the measure of center found by dividing the

measure of center: measures of center or

median: when the numbers are arranged from

mode: the number that appears most frequently in

model: to represent a mathematical situation with

multiple: the product of a whole number and any

nonlinear (function) equation: a function

non-standard units: measuring units such as

pre-image: the original figure in a transformation.

properties of 1-2- and 3- dimensional

shapes: common features of 1-, 2-, and 3-

range: the difference between the greatest and

reflection/flips: a transformation in which a figure

representations: physical objects, drawings,

rotation/turn: a transformation that forms an

rotational symmetry: a property that allows a

standard units of measure: measurements

stem- and- leaf plot: a method of

symbolic rules: rules that use variables and

translation/slide: a transformation in which an

visual model: models such as networks that

Grade Forth – California State Standards Taught

By the end of grade four, students understand large numbers and addition,

subtraction, multiplication, and division of whole numbers. They describe

and compare simple fractions and decimals. They understand the properties of,

and the relationships between, plane geometric figures. They collect, represent,

and analyze data to answer questions.

Number Sense

1.0 Students understand the place value of whole numbers and decimals to two

decimal places and how whole numbers and decimals relate to simple fractions.

Students use the concepts of negative numbers:

1.1 Read and write whole numbers in the millions.

1.2 Order and compare whole numbers and decimals to two decimal places.

1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand,

ten thousand, or hundred thousand.

1.4 Decide when a rounded solution is called for and explain why such a solution may

be appropriate.

1.5 Explain different interpretations of fractions, for example, parts of a whole, parts

of a set, and division of whole numbers by whole numbers; explain equivalents

of fractions (see Standard 4.0).

1.6 Write tenths and hundredths in decimal and fraction notations and know the

fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50;

7/4 = 1 3/4 = 1.75).

1.7 Write the fraction represented by a drawing of parts of a figure; represent a given

fraction by using drawings; and relate a fraction to a simple decimal on a number

line.

1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature,

in 뱋wing?.

1.9 Identify on a number line the relative position of positive fractions, positive mixed

numbers, and positive decimals to two decimal places.

15

2.0 Students extend their use and understanding of whole numbers to the

addition and subtraction of simple decimals:

2.1 Estimate and compute the sum or difference of whole numbers and positive

decimals to two places.

2.2 Round two-place decimals to one decimal or the nearest whole number and judge

the reasonableness of the rounded answer.

3.0 Students solve problems involving addition, subtraction, multiplication,

and division of whole numbers and understand the relationships among

the operations:

3.1 Demonstrate an understanding of, and the ability to use, standard algorithms

for the addition and subtraction of multidigit numbers.

3.2 Demonstrate an understanding of, and the ability to use, standard algorithms

for multiplying a multidigit number by a two-digit number and for dividing a

multidigit number by a one-digit number; use relationships between them to

simplify computations and to check results.

3.3 Solve problems involving multiplication of multidigit numbers by two-digit

numbers.

3.4 Solve problems involving division of multidigit numbers by one-digit numbers.

4.0 Students know how to factor small whole numbers:

4.1 Understand that many whole numbers break down in different ways

(e.g., 12 = 4 ?3 = 2 ?6 = 2 ?2 ?3).

4.2 Know that numbers such as 2, 3, 5, 7, and 11 do not have any factors except 1 and

themselves and that such numbers are called prime numbers.

Algebra and Functions

1.0 Students use and interpret variables, mathematical symbols, and properties to

write and simplify expressions and sentences:

1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions

or equations (e.g., demonstrate an understanding and the use of the concept of a

variable).

1.2 Interpret and evaluate mathematical expressions that now use parentheses.

1.3 Use parentheses to indicate which operation to perform first when writing expressions

containing more than two terms and different operations.

1.4 Use and interpret formulas (e.g., area = length ?width or A = lw) to answer

questions about quantities and their relationships.

1.5 Understand that an equation such as y = 3x + 5 is a prescription for determining

a second number when a first number is given.

2.0 Students know how to manipulate equations:

2.1 Know and understand that equals added to equals are equal.

2.2 Know and understand that equals multiplied by equals are equal.

Measurement and Geometry

1.0 Students understand perimeter and area:

1.1 Measure the area of rectangular shapes by using appropriate units, such as square

centimeter (cm2), square meter (m 2), square kilometer (km 2), square inch (in 2),

square yard (yd2), or square mile (mi 2).

1.2 Recognize that rectangles that have the same area can have different perimeters.

1.3 Understand that rectangles that have the same perimeter can have different areas.

1.4 Understand and use formulas to solve problems involving perimeters and areas

of rectangles and squares. Use those formulas to find the areas of more complex

figures by dividing the figures into basic shapes.

2.0 Students use two-dimensional coordinate grids to represent points and graph

lines and simple figures:

2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw

10 points on the graph of the equation y = 3x and connect them by using a straight

line).

2.2 Understand that the length of a horizontal line segment equals the difference of the

x-coordinates.

2.3 Understand that the length of a vertical line segment equals the difference of the ycoordinates.

3.0 Students demonstrate an understanding of plane and solid geometric objects

and use this knowledge to show relationships and solve problems:

3.1 Identify lines that are parallel and perpendicular.

3.2 Identify the radius and diameter of a circle.

3.3 Identify congruent figures.

3.4 Identify figures that have bilateral and rotational symmetry.

3.5 Know the definitions of a right angle, an acute angle, and an obtuse angle. Understand

that 90, 180, 270, and 360?are associated, respectively, with 1/4, 1/2, 3/4, and

full turns.

3.6 Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in

terms of the number and shape of faces, edges, and vertices; interpret two-dimensional

representations of three-dimensional objects; and draw patterns (of faces) for

a solid that, when cut and folded, will make a model of the solid.

3.7 Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and

identify their attributes.

3.8 Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle,

parallelogram, trapezoid).

Statistics, Data Analysis, and Probability

1.0 Students organize, represent, and interpret numerical and categorical data and

clearly communicate their findings:

1.1 Formulate survey questions; systematically collect and represent data on a number

line; and coordinate graphs, tables, and charts.

1.2 Identify the mode(s) for sets of categorical data and the mode(s), median, and any

apparent outliers for numerical data sets.

1.3 Interpret one- and two-variable data graphs to answer questions about a situation.

2.0 Students make predictions for simple probability situations:

2.1 Represent all possible outcomes for a simple probability situation in an organized

way (e.g., tables, grids, tree diagrams).

2.2 Express outcomes of experimental probability situations verbally and numerically

(e.g., 3 out of 4; 3/4).

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant from

irrelevant information, sequencing and prioritizing information, and observing

patterns.

1.2 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex problems.

2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables,

diagrams, and models, to explain mathematical reasoning.

2.4 Express the solution clearly and logically by using the appropriate mathematical

notation and terms and clear language; support solutions with evidence in both

verbal and symbolic work.

2.5 Indicate the relative advantages of exact and approximate solutions to problems

and give answers to a specified degree of accuracy.

2.6 Make precise calculations and check the validity of the results from the context

of the problem.

3.0 Students move beyond a particular problem by generalizing to other

situations:

3.1 Evaluate the reasonableness of the solution in the context of the original situation.

3.2 Note the method of deriving the solution and demonstrate a conceptual under

standing of the derivation by solving similar problems.

3.3 Develop generalizations of the results obtained and apply them in other

circumstances.

Grade Five Math – California State Standards Taught

By the end of grade five, students increase their facility with the four basic

arithmetic operations applied to fractions, decimals, and positive and negative

numbers. They know and use common measuring units to determine length and

area and know and use formulas to determine the volume of simple geometric

figures. Students know the concept of angle measurement and use a protractor

and compass to solve problems. They use grids, tables, graphs, and charts to

record and analyze data.

Number Sense

1.0 Students compute with very large and very small numbers, positive integers,

decimals, and fractions and understand the relationship between decimals,

fractions, and percents. They understand the relative magnitudes of numbers:

1.1 Estimate, round, and manipulate very large (e.g., millions) and very small

(e.g., thousandths) numbers.

1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for

common fractions and explain why they represent the same value; compute a given

percent of a whole number.

1.3 Understand and compute positive integer powers of nonnegative integers; compute

examples as repeated multiplication.

1.4 Determine the prime factors of all numbers through 50 and write the numbers as

the product of their prime factors by using exponents to show multiples of a factor

(e.g., 24 = 2 ?2 ?2 ?3 = 23 ?3).

1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and

positive and negative integers.

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2.0 Students perform calculations and solve problems involving addition, subtraction,

and simple multiplication and division of fractions and decimals:

2.1 Add, subtract, multiply, and divide with decimals; add with negative integers;

subtract positive integers from negative integers; and verify the reasonableness of

the results.

2.2 Demonstrate proficiency with division, including division with positive decimals

and long division with multidigit divisors.

2.3 Solve simple problems, including ones arising in concrete situations, involving the

addition and subtraction of fractions and mixed numbers (like and unlike denominators

of 20 or less), and express answers in the simplest form.

2.4 Understand the concept of multiplication and division of fractions.

2.5 Compute and perform simple multiplication and division of fractions and apply

these procedures to solving problems.

Algebra and Functions

1.0 Students use variables in simple expressions, compute the value of the expression

for specific values of the variable, and plot and interpret the results:

1.1 Use information taken from a graph or equation to answer questions about a

problem situation.

1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic

expressions in one variable by substitution.

1.3 Know and use the distributive property in equations and expressions with

variables.

1.4 Identify and graph ordered pairs in the four quadrants of the coordinate plane.

1.5 Solve problems involving linear functions with integer values; write the equation;

and graph the resulting ordered pairs of integers on a grid.

Measurement and Geometry

1.0 Students understand and compute the volumes and areas of simple objects:

1.1 Derive and use the formula for the area of a triangle and of a parallelogram by

comparing it with the formula for the area of a rectangle (i.e., two of the same

triangles make a parallelogram with twice the area; a parallelogram is compared

with a rectangle of the same area by cutting and pasting a right triangle on the

parallelogram).

1.2 Construct a cube and rectangular box from two-dimensional patterns and use

these patterns to compute the surface area for these objects.

1.3 Understand the concept of volume and use the appropriate units in common

measuring systems (i.e., cubic centimeter [cm 3], cubic meter [m3], cubic inch

[in 3], cubic yard [yd3]) to compute the volume of rectangular solids.

1.4 Differentiate between, and use appropriate units of measures for, two- and

three-dimensional objects (i.e., find the perimeter, area, volume).

2.0 Students identify, describe, and classify the properties of, and the relationships

between, plane and solid geometric figures:

2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles,

and triangles by using appropriate tools (e.g., straightedge, ruler, compass,

protractor, drawing software).

2.2 Know that the sum of the angles of any triangle is 180?and the sum of the angles

of any quadrilateral is 360?and use this information to solve problems.

2.3 Visualize and draw two-dimensional views of three-dimensional objects made

from rectangular solids.

Statistics, Data Analysis, and Probability

1.0 Students display, analyze, compare, and interpret different data sets, including

data sets of different sizes:

1.1 Know the concepts of mean, median, and mode; compute and compare simple

examples to show that they may differ.

1.2 Organize and display single-variable data in appropriate graphs and representations

(e.g., histogram, circle graphs) and explain which types of graphs are appropriate

for various data sets.

1.3 Use fractions and percentages to compare data sets of different sizes.

1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data

in terms of the situation depicted by the graph.

1.5 Know how to write ordered pairs correctly; for example, (x, y).

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant from

irrelevant information, sequencing and prioritizing information, and observing

patterns.

1.2 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex problems.

2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables,

diagrams, and models, to explain mathematical reasoning.

2.4 Express the solution clearly and logically by using the appropriate mathematical

notation and terms and clear language; support solutions with evidence in both

verbal and symbolic work.

2.5 Indicate the relative advantages of exact and approximate solutions to problems

and give answers to a specified degree of accuracy.

2.6 Make precise calculations and check the validity of the results from the context

of the problem.

3.0 Students move beyond a particular problem by generalizing to other

situations:

3.1 Evaluate the reasonableness of the solution in the context of the original situation.

3.2 Note the method of deriving the solution and demonstrate a conceptual under

standing of the derivation by solving similar problems.

3.3 Develop generalizations of the results obtained and apply them in other

circumstances.

Grade Six – California State Standards Taught

By the end of grade six, students have mastered the four arithmetic operations

with whole numbers, positive fractions, positive decimals, and positive and

negative integers; they accurately compute and solve problems. They apply their

knowledge to statistics and probability. Students understand the concepts of

mean, median, and mode of data sets and how to calculate the range. They

analyze data and sampling processes for possible bias and misleading conclusions;

they use addition and multiplication of fractions routinely to calculate the

probabilities for compound events. Students conceptually understand and work

with ratios and proportions; they compute percentages (e.g., tax, tips, interest).

Students know about p and the formulas for the circumference and area of a

circle. They use letters for numbers in formulas involving geometric shapes and

in ratios to represent an unknown part of an expression. They solve one-step

linear equations.

Number Sense

1.0 Students compare and order positive and negative fractions, decimals, and

mixed numbers. Students solve problems involving fractions, ratios, proportions,

and percentages:

1.1 Compare and order positive and negative fractions, decimals, and mixed numbers

and place them on a number line.

1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour)

to show the relative sizes of two quantities, using appropriate notations (a/b, a to b,

a:b).

1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21,

find the length of a side of a polygon similar to a known polygon). Use crossmultiplication

as a method for solving such problems, understanding it as the

multiplication of both sides of an equation by a multiplicative inverse.

1.4 Calculate given percentages of quantities and solve problems involving discounts

at sales, interest earned, and tips.

24

2.0 Students calculate and solve problems involving addition, subtraction,

multiplication, and division:

2.1 Solve problems involving addition, subtraction, multiplication, and division of

positive fractions and explain why a particular operation was used for a given

situation.

2.2 Explain the meaning of multiplication and division of positive fractions and per8

÷ 15

8 × 16 form the calculations (e.g., 5. .16 = 5. .15 = 2.3).

2.3 Solve addition, subtraction, multiplication, and division problems, including those

arising in concrete situations, that use positive and negative integers and combinations

of these operations.

2.4 Determine the least common multiple and the greatest common divisor of whole

numbers; use them to solve problems with fractions (e.g., to find a common

denominator to add two fractions or to find the reduced form for a fraction).

Algebra and Functions

1.0 Students write verbal expressions and sentences as algebraic expressions and

equations; they evaluate algebraic expressions, solve simple linear equations,

and graph and interpret their results:

1.1 Write and solve one-step linear equations in one variable.

1.2 Write and evaluate an algebraic expression for a given situation, using up to three

variables.

1.3 Apply algebraic order of operations and the commutative, associative, and distributive

properties to evaluate expressions; and justify each step in the process.

1.4 Solve problems manually by using the correct order of operations or by using a

scientific calculator.

2.0 Students analyze and use tables, graphs, and rules to solve problems involving

rates and proportions:

2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters

to inches).

2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value

of another quantity.

2.3 Solve problems involving rates, average speed, distance, and time.

1

3.0 Students investigate geometric patterns and describe them algebraically:

3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l,

A =.2 bh, C = π d.the formulas for the perimeter of a rectangle, the area of a triangle,

and the circumference of a circle, respectively).

3.2 Express in symbolic form simple relationships arising from geometry.

Measurement and Geometry

1.0 Students deepen their understanding of the measurement of plane and solid

shapes and use this understanding to solve problems:

1.1 Understand the concept of a constant such as π; know the formulas for the circumference

and area of a circle.

1.2 Know common estimates of π (3.14; 22.7) and use these values to estimate and calculate

the circumference and the area of circles; compare with actual measurements.

1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area

of base × height); compare these formulas and explain the similarity between them

and the formula for the volume of a rectangular solid.

2.0 Students identify and describe the properties of two-dimensional figures:

2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide

descriptions of these terms.

2.2 Use the properties of complementary and supplementary angles and the sum of the

angles of a triangle to solve problems involving an unknown angle.

2.3 Draw quadrilaterals and triangles from given information about them (e.g., a

quadrilateral having equal sides but no right angles, a right isosceles triangle).

Statistics, Data Analysis, and Probability

1.0 Students compute and analyze statistical measurements for data sets:

1.1 Compute the range, mean, median, and mode of data sets.

1.2 Understand how additional data added to data sets may affect these computations

of measures of central tendency.

1.3 Understand how the inclusion or exclusion of outliers affects measures of central

tendency.

1.4 Know why a specific measure of central tendency (mean, median, mode) provides

the most useful information in a given context.

2.0 Students use data samples of a population and describe the characteristics

and limitations of the samples:

2.1 Compare different samples of a population with the data from the entire population

and identify a situation in which it makes sense to use a sample.

2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses

to a survey, random sampling) and which method makes a sample more representative

for a population.

2.3 Analyze data displays and explain why the way in which the question was asked

might have influenced the results obtained and why the way in which the results

were displayed might have influenced the conclusions reached.

2.4 Identify data that represent sampling errors and explain why the sample (and the

display) might be biased.

2.5 Identify claims based on statistical data and, in simple cases, evaluate the validity

of the claims.

3.0 Students determine theoretical and experimental probabilities and use these

to make predictions about events:

3.1 Represent all possible outcomes for compound events in an organized way

(e.g., tables, grids, tree diagrams) and express the theoretical probability of each

outcome.

3.2 Use data to estimate the probability of future events (e.g., batting averages or

number of accidents per mile driven).

3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and

percentages between 0 and 100 and verify that the probabilities computed are

reasonable; know that if P is the probability of an event, 1-P is the probability of an

event not occurring.

3.4 Understand that the probability of either of two disjoint events occurring is the sum

of the two individual probabilities and that the probability of one event following

another, in independent trials, is the product of the two probabilities.

3.5 Understand the difference between independent and dependent events.

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant from

irrelevant information, identifying missing information, sequencing and

prioritizing information, and observing patterns.

1.2 Formulate and justify mathematical conjectures based on a general description

of the mathematical question or problem posed.

1.3 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex problems.

2.3 Estimate unknown quantities graphically and solve for them by using logical

reasoning and arithmetic and algebraic techniques.

2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables,

diagrams, and models, to explain mathematical reasoning.

2.5 Express the solution clearly and logically by using the appropriate mathematical

notation and terms and clear language; support solutions with evidence in both

verbal and symbolic work.

2.6 Indicate the relative advantages of exact and approximate solutions to problems

and give answers to a specified degree of accuracy.

2.7 Make precise calculations and check the validity of the results from the context

of the problem.

3.0 Students move beyond a particular problem by generalizing to other

situations:

3.1 Evaluate the reasonableness of the solution in the context of the original situation.

3.2 Note the method of deriving the solution and demonstrate a conceptual under

standing of the derivation by solving similar problems.

3.3 Develop generalizations of the results obtained and the strategies used and apply

them in new problem situations.

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